Transition for Optimal Paths in Bimodal Directed Polymers

The problem for optimal paths in bimodal directed polymers is studied. It is shown that the distribution of the thermal average position of the endpoints of the optimal paths is discontinuous below the threshold p
c. The origin is that there is a finite possibility that only one endpoint takes the global minimum energy for p
c. Our results suggest that the percolation threshold for directed percolation is also the critical point of the transition for the possibility that the optimal paths converge to one endpoint.